Introduction to Modern Geometry

Modern Geometry:

The Directed Line Segment

In Euclidean Geometry, line segments possess only magnitude, and not direction.

Figure 1: In Euclidean Geometry, only the magnitude of line segments is considered.

As we can see in Figure 1, in Euclidean Geometry, the line segment:

|AB|

would be considered equal to the line segment:

|BA|

However, this is not the case with Modern Geometry.

In Modern Geometry, line segments not only possess magnitude, but direction also.

Figure 2: In Modern Geometry, line segments possess direction as well as magnitude.

In Modern Geometry:

|AB|

is termed:

‘a directed line segment’

Figure 3: In Modern Geometry, we may say that we have inverted the direction of the line segment as found in Figure 3:.

In Modern Geometry, we would call the line segment, as depicted in Figure 3:

|BA|

In Modern Geometry, as regards direction:

|AB| ≠ |BA|

Because, in Figure 3, we have merely reversed the direction of the line-segment:

|AB|

, then we may also call line-segment:

|BA|

by the equivalent:

- |AB|

Therefore, as regards direction, in Modern Geometry:

|AB| = |-BA|

and, also:

|BA| = |-AB|